Notch sensitivity and short cracks tolerance in a super 13Cr stainless steel under sulfide stress corrosion cracking conditions

1 Introduction

Sulfide stress corrosion cracking (SSC) failures can cause major problems in practice, in particular in the oil and gas industry (Craig 2003; Jones 1992; Kane 1998; Korb and Olson 1992). Such failures might occur under static loads even in structural components made of corrosion resistant alloys (CRA), like the weldable UNS S41426 martensitic stainless steel, also known as S13Cr, studied here. This 13Cr5Ni2Mo quenched and tempered steel can be supplied with a minimum yield strength S_Y = 760 MPa (110 ksi) and yet be tough enough for severe offshore tubing and casing applications, including well service in corrosive environments containing CO₂, H₂S (typically up to 10 kPa or 1.5 psi partial pressure of H₂S), and high chloride content.

It is well known that sulfide hydrogen gas H₂S, an undesirable but not uncommon contaminant in oil wells, plays an important role on SSC failures (Craig 2003; Jones 1992; Kane 1998; Korb and Olson 1992; NACE/ISO 2015). Like any other environmentally assisted cracking (EAC) mechanism, SSC is a mechanical-chemical problem jointly induced by both tensile stresses and aggressive media. Indeed, the cause for SSC failures is the synergistic interaction between high enough tensile stresses with metallurgical features and physicochemical peculiarities of the corrosion cracking processes, such as electrochemical reactions inside the aggressive sulfide environments. Static loads that induce tensile stresses σ > S_SSC can start cracks in the component surface, where S_SSC is the material cracking resistance inside the sulfide environment. They can also support crack growth (caused by SSC) when the crack stress intensity factor (SIF) K_I > K_ISSC, the long crack growth threshold inside the same environment (Castro and Leite 2013; Castro and Meggiolaro 2013; Castro et al. 2015a, b; Craig 2003; Jones 1992; Korb and Olson 1992). This same idea also applies to any other EAC mechanism.

However, this work deals with the tolerance to short (not long) cracks that depart from surface notches, which locally concentrate stresses and enhance the crack starting tendency. In such cases, the stress gradients ahead of the notch tip much affect the short crack behavior. In fact, they are the cause for notch sensitivity effects under SSC (or any other EAC) conditions (Castro and Leite 2013; Castro and Meggiolaro 2013; Castro et al. 2015a, b). This idea is the basis for a model to quantify notch sensitivity under any EAC mechanism using sound mechanical principles based on an analogy with similar fatigue problems (Castro and Meggiolaro 2013, 2016; Castro et al. 2012; Frost 1959; Meggiolaro et al. 2007; Wu et al. 2010). Indeed, short crack tolerance predictions in notched specimens based on this chemical/mechanical model have already been verified under hydrogen embrittlement conditions in a 4140 steel immersed in the NACE TM0177 B solution, as well as under liquid metal embrittlement in an Al-Ga pair (Castro et al. 2015b).

The definition for notch sensitivity in EAC, 0 ≤ q_EAC ≤ 1, is analogous to its definition in fatigue:

where K_t is the linear elastic (LE) stress concentration factor (SCF) induced by the notch.

This definition implies that, albeit the maximum LE stress at notch tips is σ = K_t⋅σ_n, where σ_n is the nominal stress (which neglects notch-induced stress concentration effects) applied on the notched component, the actual notch-induced strength reduction factor under EAC conditions is K_tEAC ≤ K_t. This means that the component can support a local stress up to σ = S_EAC/K_tEAC > S_EAC/K_t at its notch tip without failing. Indeed, albeit localized stresses σ > S_EAC/K_t can induce tiny cracks at notch tips, they stop after growing for a while under a nominal stresses σ_n < S_EAC/K_tEAC. This phenomenon has been observed and related to notch sensitivity in fatigue by Frost (1959), and extended to EAC by Castro and Meggiolaro (2013), Castro and Leite (2013), and Castro et al. (2015a). The reason why cracks can start from notch tips and then stop after growing for a while is the competition between the increase in the SIF K_I due to the crack size increment, and the decrease in K_I due to steep stress gradients ahead of notch tips. Indeed, SIFs for such cracks are given by K_I ≈ 1.12⋅K_gr⋅σ√(πa), where 1.12 is the factor associated to their free surface and K_gr is their stress gradient factor (Miranda et al. 2019). The SIF increase caused by the growing crack lengths a can thus be compensated by the decrease in the local stress σ ahead of their tips induced by the notch-induced stress gradients, leading the cracks to stop when their K_I < K_IEAC(a). K_IEAC(a) is the short crack growth threshold in EAC, which is crack-size dependent, as explained in the following.

Before using these ideas to model the notch sensitivity factor q_EAC, it is necessary to emphasize that short cracks cannot behave like long cracks. Indeed, it is well known that the driving force for cracks is the applied SIF K_I ≈ σ√(πa) when they are loaded under predominantly linear elastic (LE) conditions (see e.g. Meggiolaro et al. 2007; Wu et al. 2010; Castro et al. 2012; Castro and Meggiolaro 2013, 2016; and Miranda et al. 2019 for details). Moreover, if fracture is induced when K_I = K_IC, it is non-sense to assume that if a → 0 it would be necessary to have a stress σ → ∞ to cause it. This fact was recognized by Topper et al. a long time ago in fatigue, who claimed that if a → 0 then σ → S_L, the fatigue limit of the material (El Haddad et al. 1979a,b; Kitagawa and Takahashi 1976). This same idea can be applied to EAC problems, providing a solid mechanical basis for new design procedures in this field. These procedures can be quite attractive from an economical perspective, in view of the very high costs associated with the overly simplified pass/fail criteria used nowadays.

Indeed, for structural analysis and design purposes, EAC problems usually are treated by an over-conservative policy on susceptible material-environment pairs: if aggressive media are unavoidable during service lives of sensitive structural components, the standard solution is to build them from a material immune to EAC in those media. A similar but less expensive alternative is to protect the component surface with a suitable coating, if such a coating is available. EAC-proof coatings must be strongly adherent, scratch resistant, and much more reliable than common coatings used in general for corrosion protection, because structural components can fail without warning under EAC.

Albeit over-conservative design criteria may be a sensible way to avoid failures, they can also be unjustifiable economically if an equally strong and tough but less costly material is disqualified only because it might suffer EAC in the service environment, without considering any stress analysis issues. Moreover, such pass/fail criteria cannot be used for structural integrity evaluations when the media surrounding the component changes during its operational life, for instance when benign conditions in oil and gas wells deteriorate because e.g. bacteria growth starts to produce H₂S in them.

In fact, EAC damage simply cannot be properly evaluated neglecting the influence of the stress fields that drive them. These must of course include both the stresses induced by service loads and the residual stresses eventually caused by previous loads and overloads, or else by maintenance or manufacturing processes. Even though it may be difficult to precisely identify and quantify EAC conditions in practical applications, due to the many variables that may affect such problems, chemical and metallurgical tools alone are not enough to evaluate them, since such conditions depend on mechanical parameters too. Thus, serious structural integrity evaluations must include stress analyses to estimate maximum tolerable flaw sizes in EAC problems. Such chemical/mechanical techniques should be mandatory in the design stage, but the usual practice still is to use intrinsically safe, albeit usually too expensive, pass/fail criteria proposed in many design codes to substitute them.

However, although unusual in the design stage, proper joint chemical/mechanical analyses are simply unavoidable when evaluating the structural integrity of components not originally designed for EAC service, when by any reason they begin to work under such conditions due to unpredicted operational changes. For instance, consider a common steel pipeline that after some time in service begins to transport originally unforeseen amounts of H₂S, due to changes in the oil well conditions. In most practical cases, profit losses associated with the very long time required to replace a pipeline can be simply too high to be ignored, especially in offshore applications. Hence, albeit the original pipeline steel may be EAC-sensitive, it may be unavoidable to take the structural risk of maintaining it in service while a new H₂S-resistant line is designed, built, and commissioned. In such cases, chemical/mechanical analyses may be the only way to access risk levels associated with maintaining the original pipeline in service.

Indeed, such risky decisions can be managed in principle by the methodology developed in this work, which introduces chemical concepts to extend to EAC problems the mechanical analysis developed to model notch sensitivity effects in fatigue (Castro and Meggiolaro 2013, 2016; Castro et al. 2012; Miranda et al. 2019; Meggiolaro et al. 2007; Wu et al. 2010).

2 Short crack tolerance under EAC conditions

Like in the fatigue case, tolerable short cracks cause the difference between K_t and their actual effect K_tEAC ≤ K_t in the strength of notched components under EAC conditions. Hence, if cracks behave well under EAC, as they behave under fatigue conditions, i.e. if SIFs can be used to describe them, following El Haddad et al. (1979a,b) (ETS) ideas to model the Kitagawa and Takahashi (1976) diagram trend, a “short-crack characteristic size under EAC conditions” for the Griffith plate can be proposed as:

where S_EAC and K_IEAC are, respectively, the material resistances to crack initiation and to crack growth under EAC inside the aggressive environment.

This simple idea assumes all chemical effects that may affect the EAC behavior in any given environment-material pair can be quantified properly by S_EAC and by K_IEAC. These are well-defined mechanical properties, measurable under fixed loads inside the aggressive environment by standard procedures (ASTM International 2009, 2013a,b, 2021; NACE International 2016a,b). Moreover, albeit EAC problems are time-dependent, S_EAC and K_IEAC are not, since they quantify limit stresses or SIFs required to start or to pursue the EAC processes. Hence, assuming S_EAC and K_IEAC are analogous to the equivalent fatigue limit ΔS_L(R) and fatigue crack growth (FCG) threshold ΔK_th(R) parameters, then a Kitagawa-Takahashi-like diagram can be proposed to quantify the crack sizes a tolerable by any structural component that works in EAC conditions under a given tensile stress σ. The ranges ΔS_L(R) and ΔK_th(R) depend on R = σ_min/σ_max because fatigue has two driving forces, σ_max and Δσ for crack initiation, and K_max and ΔK for FCG. The ranges Δσ and ΔK drive cyclic plasticity-induced fatigue damage mechanisms, while σ_max and K_max drive static damage mechanisms, such as fracture and EAC, as discussed elsewhere (Castro and Meggiolaro 2016).

Since SIFs are in general given by K_I = σ√(πa)⋅g(a/w), Yu et al. (1988) used the geometry factor g(a/w) to generalize the fatigue equivalent of the ETS model. Following this idea, the short-crack characteristic size under EAC can be redefined for any geometry by:

The largest stress that does not propagate microcracks by EAC in this case is also S_EAC, as it should be: if a << a_0EAC, then K_I = K_IEAC⇒ σ → S_EAC. However, if the crack initiates from a notch, as usual, its local driving force σ is the stress at the notch tip, not the nominal stress usually used in SIF expressions. Since in these cases the g(a/w) factor usually includes the notch stress concentration effect, it is better to split it into two parts, through g(a/w) = η⋅K_gr(a). The right factor K_gr(a) quantifies the stress gradient effect near the notch tip, which for microcracks tends to K_t at the notch tip, i.e. K_gr(a → 0) → K_t, and for long cracks tends to 1, i.e. K_gr(a >> a₀) → 1 (Miranda et al. 2019). On the other hand, the constant η from the left factor of g(a/w) quantifies the notch free surface effect, which also affects K_I, in which case η = 1.12. As a result, it is better to redefine a_0EAC by:

In fact, as the stress at the notch tip must be smaller than the material resistance to EAC to avoid cracking, i.e. σ(a → 0) = K_t⋅σ_n = K_gr(0)⋅σ_n < S_EAC, the stress gradient given by K_gr(a) does not play a role in the a_0EAC expression. However, since K_I is a crack driving force, it should be material-independent. Thus, the a_0EAC effect on the short-crack behavior should be used to modify the EAC threshold instead of the SIF K_I, making it a function of the crack size and of the EAC limits, a quite convenient assumption from the operational point of view. This way, the a-dependent EAC threshold K_IEAC(a) becomes

It may also be convenient to assume Equation (5) is just one of the models that obey the long-crack and the microcrack limit behaviors, introducing in it a data fitting parameter γ proposed by Bazant (1997) for the fatigue case, to obtain:

This equation reproduces Equation (5) when γ = 2, as well as the bilinear limits for γ → ∞ shown in Figure 1, namely σ = S_EAC if a < a_0EAC and σ = K_IEAC/√(πa) if a > a_0EAC. This additional parameter may allow Equation (6) to fit better experimental data, but the ETS γ = 2 value should be used if no further information is available. The curves shown in Figure 1 illustrate the influence of the γ data-fitting parameter on the minimum stress σ(a) needed to propagate short or long cracks under EAC for a given crack size a:

Figure 1:

Modified Kitagawa-Takahashi diagram to describe the behavior of short and long cracks for structural design purposes, plotting the stress σ needed for propagating by EAC any crack of size a (the two limit conditions σ = S_EAC and σ = K_IEAC/√(πa)g(a/w) cross at a_0EAC).

Since σ(a) is the minimum stress needed to propagate cracks of any size a by EAC, the curve σ(a) limits the domain on the σ × a plane where non-propagating cracks can exist in any given material-environment pair. Figure 1 also shows how this tolerable short crack curve tends to the bilinear growth limit bounded by σ = S_EAC (the crack initiation limit) for a → 0, and by σ = K_IEAC/√(πa) (the crack growth threshold for long cracks, with size a >> a_0EAC). Notice that the larger the γ value is, the faster the short crack behavior tends to that bilinear EAC limit.

This idea can be used to propose as well a generalized Kitagawa-Takahashi diagram with four regions that may contain non-propagating cracks (Castro and Leite 2013; Vasudevan and Sadananda 2011), see Figure 2. First, the lower region bounded by ΔS_L(R) and ΔK_th(R)/√(πa), which limit the material tolerance to non-propagating fatigue cracks under constant amplitude loads in any given non-inert environment. Second, the region bounded by S_EAC and K_IEAC/√(πa), which may contain non-propagating EAC cracks under fixed static loads in that environment. Third, the region bounded by ΔS_Lvac and ΔK_thvac, the R-independent fatigue limit and FCG threshold of that material in vacuum, which limits its intrinsic resistance to non-propagating fatigue cracks. Finally, the region limited by the intrinsic material properties S_Uvac and K_ICvac/√(πa), which can only be measured in vacuum or in truly inert environments. The advantage of looking at the cracking problem in such an integrated way is that this approach allows the modeling of both mechanical and chemical damage by a single or unified analysis methodology.

Figure 2:

Generalized Kitagawa-Takahashi diagram showing the stresses (including the residual ones) σ and the stress ranges Δσ required for crack growth under EAC and fatigue, considering the strength reductions caused by the environment-material chemistry.

Hence, assuming the driving force for cracks loaded under EAC is indeed the SIF applied on them, and the chemical effects that affect their behavior can be quantified by the material resistances to crack initiation S_EAC and to crack growth K_IEAC, then cracks induced by EAC may depart from notches and then stop, eventually becoming non-propagating. Notch-induced stress gradient effects on the SIFs K_I(a) of short cracks that depart from notch tips are the primary cause for this behavior, as explained above. In such cases, the limit size of non-propagating short cracks can be evaluated by procedures similar to those used in fatigue, and the tolerance to such defects can be quantified by a specific notch sensitivity factor for EAC in structural integrity assessments. Hence, for any crack size a, the criterion for the maximum tolerable stress under EAC conditions is given by:

These equations allow explicit stress analyses of notches under EAC conditions for structural design purposes. Hence, they can replace the pass/fail criterion used to “solve” most practical EAC problems nowadays. In fact, they are a sound chemical/mechanical criterion for EAC that can be used by structural engineers without requiring much expertise in chemistry. Moreover, it can be verified experimentally, as illustrated in the following sections. Notice that this claim by no means implies that the mechanical approach proposed here neglects the chemical aspects of the EAC mechanism. On the contrary, if the important chemical effects on EAC can be quantified properly by S_EAC and K_IEAC alone then, and only then, Equation (8) can be useful. If they cannot, then this analysis is simplistic and must be improved to consider time-dependent cracking mechanisms. An example of that is when the aggressive environment induces joint pitting and EAC, since in such cases the stress concentration factors at pit tips keep changing along the time.

3 Engineering estimates for notch effects on the FCG behavior of short cracks

According to Inglis (1913), notch effects in general can be approximated by replacing them with an ellipse with tip radius ρ = c²/b tangent to their tips, where b and c are the ellipse semi-axes. If in an Inglis’ plate the ellipse 2b axis is centered at the x-axis origin and is perpendicular to the nominal stress σ_n, then the stress concentration gradient f₁ ahead of the ellipse notch tip is given by:

The peculiar growth/stop behavior of short cracks that initiate from sharp notches is easy to justify by the high stress gradient ahead of elongated elliptical hole tips. Indeed, the LE stress concentration effects produced by any elliptical hole with b ≥ c fall sharply from K_t = σ_y(1, 0)/σ_n ≥ 3 at its tip down to 1.82 < K_1.2 = K_t(x/b = 1.2) = σ_y(1.2, 0)/σ_n < 2.11 at the point located just b/5 ahead of it. Hence, at such points σ_y(1.2, 0)/σ_n ≅ 2, independently of the elliptical notch K_t, see Figure 3 (Castro and Meggiolaro 2016).

Figure 3:

The ratio K_1.2 = σ_y(x/b = 1.2, 0)/σ_n at the point that is just b/5 ahead of the tip of any elliptical hole is almost independent of its K_t in the LE case (mind the scale).

The SIFs K_I ≅ 1.12σ√(πa)⋅f₁(K_t, a) of short cracks that depart from sharp notch tips first increase fast with the growing crack size, but for large K_t values they can even decrease as the short cracks grow, until their SIF K_I eventually can start to grow again with a for longer crack sizes. Indeed, the term √a that tends to increase the SIF K_I can be overcome by the abrupt fall in the (local) stress σ due to sharp stress gradients f₁ ahead of the notch tips. In other words, such SIFs can have large but fast decreasing initial derivatives, which can even become negative for elongated notches with large K_t, see Figure 4. Thus, short fatigue cracks that initiate from these notch tips can even stop after growing for a small distance, if their SIFs eventually become smaller than the a-dependent EAC threshold. Moreover, the behavior of such non-propagating cracks can be evaluated by K_I(a) and K_IEAC(a) estimates, as clarified by two simple examples, described as follows.

Figure 4:

Stress gradient effects ahead of notch tips on the SIFs estimated for small cracks a ≤ b/5 that start from the tip of Inglis holes with b = 10 mm and various K_t = 1 + 2b/c.

Assume that a large plate with yield strength S_Y = 1200 MPa and EAC resistances S_EAC = 200 MPa and K_IEAC = 9 MPa√m, works under a nominal tensile stress σ_n = 50 MPa. First, let’s check whether it is possible to insert a circular central hole with diameter d = 20 mm without leading this plate to fail by EAC. After that, repeat the same analysis with an elongated elliptical hole with axes 2b = 20 mm (perpendicular to σ_n) and 2c = 2 mm, instead of a circular hole.

In the first example, the circular hole has a safety factor ϕ_EAC = S_EAC/(K_t⋅σ) = 200/150 ≅ 1.33 against crack initiation by EAC, so it is safe for this plate. The elliptical hole from the second example, on the other hand, has a small tip radius ρ = c²/b = 0.1 mm and K_t = 1 + 2b/c = 21, thus it would work under S_EAC = 200 < σ = K_t ⋅σ_n = 1050 MPa < S_Y = 1200 MPa at its tip. Hence, albeit its tip does not yield, its high local stress should initiate a crack by EAC and would not be admissible by traditional design procedures. However, it is worthwhile to reevaluate this prediction considering how the stress gradient effects ahead of the notch tip affect the growth of a short crack that starts from it.

The SIF estimate for a short crack at the elliptical notch tip K_I ≅ 1.12⋅σ_n√(πa)⋅f₁. However, for the circular hole, the simpler Kirsch’s solution can be used instead to estimate the SIF of small cracks that start at its border by K_I ≅ 1.12⋅σ_n√(πa)⋅f₂, where f₂ = [1 + 0.5(d/2x)² + 1.5(d/2x)⁴] (Timoshenko and Goodier 1970; Castro and Meggiolaro 2016). In both equations, a is the crack size, x is the distance from the hole center, and f₁ = f₂ when b = c, as they should.

Assuming the a-dependent crack growth threshold and the short crack characteristic size under EAC are K_IEAC(a) = K_IEAC/√(1 + a_0EAC/a) and a₀ = (1/π) (K_IEAC/ηS_EAC)² = (1/π) (9/1.12⋅200)² ≅ 0.5 mm, the SIFs K_I(a) estimated for the two holes are compared to the short-crack threshold K_IEAC(a) in Figure 5. Intersections of K_I(a) with K_IEAC(a) indicate crack arrest, thus the largest tolerable crack sizes. Estimating the behavior of short cracks near the notch tip and knowing terminal failures include crack generation and growth up to the cracked piece fracture, this model predicts that both the circular and the elliptical holes could bear the nominal load σ_n = 50 MPa without failing by EAC.

Figure 5:

No crack initiates at the circular hole (which tolerates cracks up to a_tol < 1.52 mm), while the crack that initiates at the elliptical hole stops at a_st ≅ 0.33 mm.

Indeed, considering the short-crack behavior, the circular hole would resist crack initiation by EAC under σ_n = 50 MPa and could tolerate cracks at its border up to a_tol ≅ 1.52 mm. On the other hand, a crack would initiate at the elliptical hole border and grow by EAC for a while under such a fixed σ_n load, until stopping after reaching a size a_st ≅ 0.33 mm. Hence, even though this hole would cause crack initiation, it could tolerate its non-propagating crack and would not fail by EAC, if the load remains fixed during its entire operational life.

However, the elliptical hole is much less robust than the circular one, as cracks that initiate from it are quite sensitive to operational parameters. If the load, for instance, increases by just 10%, up to σ_n = 55 MPa, then the crack initiated at its border would grow and lead the plate to fracture, while the circular hole would still resist initiation and tolerate small cracks up to a ≅ 1 mm. Similarly, if the threshold was a bit smaller, say K_IEAC = 8 MPa√m, then cracks initiated at the elliptical hole border under σ_n = 50 MPa would ruin the plate, but the circular hole would still resist initiation and tolerate small cracks, once again up to a ≅ 1 mm.

This simplified analysis is clearly more powerful than simplistic pass/fail criteria, which albeit intrinsically safe can be too expensive, especially when they require equipment replacement due to changes in the environment chemistry. Anyway, it can at least provide a simple quantitative tool that allows reasonable engineering decisions in structural integrity evaluations in changing environments, untreatable by pass/fail criteria. Finally, two words of caution. The simple examples worked here are educated estimates, not precise calculations, which need more elaborate K_gr evaluations considering equilibrium requirements while the short cracks grow from the notch tip, see Miranda et al. (2019). Moreover, this model only applies to LE stress gradients. For elastoplastic problems, when the stress is high enough to yield the notch tips, they need suitable adaptations, see Machado et al. (2017).

4 Verification of the tolerance to short crack predictions under SSC conditions

UNS S41426 S13Cr weldable martensitic stainless steels with good corrosion and mechanical resistances, including low-temperature toughness, are popular choices to solve some tough H₂S and/or CO₂ problems in oil and gas pipelines. After hot rolling, the seamless tubes are heat treated by quenching and tempering. The tempering temperature is a key parameter to their final mechanical and corrosion resistances. They can contain micro-additions of N, Ti, V, and/or Nb, to increase their mechanical strengths through carbide and carbo-nitride precipitation during heat treatments (Escobar et al. 2018; Lian et al. 2015; Udod et al. 2016). Typically, their microstructure consists of tempered martensite with small amounts of interlath and intergranular residual austenite, as well as coarse TiN and fine Ti(C,N) and/or V(C,N) precipitates (Escobar et al. 2018). S13Cr steels are purchased in 650 or 760 MPa (95 or 110 ksi) classes, based on their minimum yield strength S_Ymin. However, they may suffer sulfide stress corrosion cracking in aqueous solutions with small amounts of H₂S gas at low pH, namely H₂S partial pressure greater than 10 kPa (1.5 psi) and pH < 4.5 (NACE/ISO 2015; Tavares et al. 2017). Other compounds in aqueous solutions may accelerate the SSC damage, such as oxygen, organic acids, and chloride. In addition, pressure and temperature can affect it too.

ASME, ASTM, EFC, ISO, and NACE standards and guidelines are widely used to evaluate and qualify corrosion resistant alloys (CRA) exposed to sulfide-containing environments, among them the UNS S41426 steel. Most of them use pass/fail criteria, based on suitable specimens developing or not cracks after exposed to the aggressive environment under a specified tensile stress during a long enough testing time. Furthermore, many design parameters for sour service are purely empirical. For example, ASME even recommends a maximum design stress of 0.67⋅S_Ymin for pipes (ASME 2017), and NACE/ISO (2015) recommends SSC tests under 0.9⋅S_Ymin, using proof rings (NACE International 2016b) or four-point bending tests (NACE International 2016a). However, no standard or guideline nowadays considers fracture mechanics parameters to allow proper stress analyses in the design or evaluation of structural components for use in sulfide environments capable of causing SSC.

To use the ideas studied in the previous sections, first it is assumed here that all physicochemical effects involving SSC issues can be properly described by two material properties: the resistance to crack nucleation and growth inside the aggressive environment, S_SSC and K_ISSC, respectively. These properties must be measured after a time long enough to become time-independent, i.e. they must be compatible with the kinetics of the synergistic interaction of mechanical (tensile) stress, metallurgical characteristics of the steel, and physicochemical processes. The experimental procedures used to do so are described next.

4.2 Environmental conditions

SSC was induced in suitable specimens at 23 ± 2 °C using the NACE TM177 solution C (NACE International 2016b), composed of distilled water with 100 g/L of NaCl, adjusted with hydrochloric acid HCl to obtain pH 4.0, at a pressure of 100 kPa with 25 kPa partial pressure of H₂S and 75 kPa of CO₂. This is an even more aggressive environment than the one used by Marchebois et al. (2007) to induce SSC failures in S13Cr steel specimens under σ = 0.9⋅S_Y and 10 kPa (1.5 psi) partial pressure of H₂S, see Figure 6.

Figure 6:

SSC susceptibility diagram of an S13Cr stainless steel with S_Y = 854 MPa (124 ksi) immersed in a 100 g/L NaCl solution for specimens loaded under σ = 0.9⋅S_Y (adapted from Marchebois et al. 2007, highlighting the test condition used in this work).

4.3 Experimental results

Smooth tension, DCB, and modified C(T) specimens used in the following measurements were machined from an as received UNS S41426 tube with diameter 224.5 and 13.8 mm wall thickness. The aggressive solution was chosen to induce SSC, but not pitting in the test specimens. First, acetic acid was used to adjust the solution pH, but it induced several undesirable corrosion pits in the S13Cr steel specimens in addition to the desired SSC cracks. Hence, the aggressive solution pH was adjusted with HCl instead. A previous immersion test made following ASTM G31 and G46 procedures indicated it could avoid undesirable pitting. Indeed, all UNS 41,426 specimens tested after this acid change presented a different corrosion behavior, with negligible corrosion rates and without pits, see Figure 7. Notice that pitting is undesirable because it induces random and time-dependent stress concentration factors, which are difficult to quantify except for in a statistical way. Since the objective of this paper is to verify whether the proposed model can predict short crack tolerance from stable notches under SSC conditions, any concurrent damage mechanism should be avoided because it could mask the desired verification.

Figure 7:

13Cr surface with no pits after 30 days of gravimetric test under 25.4 kPa (3.7 psi) ppH₂S, 75.8 kPa (11.0 psi) ppCO₂, pH 4.0, 25 °C, 1bar, and 100 g/L NaCl (left); and its corrosion rate over time (right).

After eliminating the pitting corrosion problem from the aggressive solution, standard tests were performed to measure S_SSC and K_ISSC, the material resistances to crack initiation and growth under SCC, the essential chemical data for the proposed model. The crack initiation threshold S_SSC was determined according to ASTM F1624 and NACE TM0177-2016 standards, using NACE TM0177-2016 tensile specimens, method A, in step-loading tests, see Figure 8. The tests were performed using a proof ring previously calibrated by a reference load cell, see Figure 9A and B, and also in a servo-mechanical testing machine, see Figure 9C. The tests adopted small 10 MPa load steps, initially with 12 h holding time at each step, followed by 24 h holding time at each step. The dissolved oxygen was removed by purging the solution and the test vessel for 2 h per liter with 99.999% pure nitrogen gas before the specimen was immersed into the chloride aqueous solution. As the test vessel must be immune to the aggressive environment, it was manufactured using glass and PTFE. After the purge, the anaerobic solution was transferred to the test vessel, and the saturation with the test gases was performed for at least 2 h. A double wall cell with constant nitrogen flow was used to ensure that the oxygen amounts remained lower than 5 ppb, and the system was checked by dissolved oxygen sensors. The S13Cr crack initiation threshold measured with this method was S_SSC = 461 ± 23 MPa.

Figure 8:

Dimensions of tensile and DCB specimens, used respectively in S_SSC and K_ISSC tests.

Figure 9:

Step loading tests: (A) loading system used to calibrate the proof ring; (B) proof ring experimental assembly; (C) test cell assembly used for tests in a servo-mechanical machine.

The crack growth threshold K_ISSC was measured testing three side-grooved Double Cantilever Beam (DCB) specimens, see Figure 8, following standard NACE TM0177-16 method D procedures. These specimens were immersed in the anaerobic aggressive solution for 30 days, loaded by double tapered wedges that induced an initial applied K_I = 45 MPa√m. The K_ISSC measurements used the same procedures adopted in S_SSC experiments to avoid oxygen contamination of the solution along the tests. After 30 days, the specimens were taken from the corrosive environment and the load associated with the specimen deflection induced by the wedges was measured in a servo-mechanical machine, loading the specimens until releasing the load applied by the wedge. Finally, the specimens were broken in liquid Nitrogen to measure the SSC crack length, using a stereo microscope. The crack growth threshold measured in this way was K_ISSC = 36.9 ± 0.6 MPa√m.

Figure 10 shows the fracture surface of one smooth tensile specimen used in the S_SSC tests, with two different regions clearly identified: one (named A in the figure) corresponding to crack initiation and propagation inside the aggressive environment, and the other (named B) to the final rupture. Fracture surfaces (A) and (B) indicate brittle fracture and plastic collapse mechanisms, respectively. At 1000x magnification, (A) presents mixed-mode, intergranular and transgranular cleavage, whereas (B) shows microvoid coalescence and dimples. Figure 11 shows the K_ISSC specimens.

Figure 10:

S _SSC S13Cr steel specimen fractography after rupture in the aggressive environment (aqueous solution containing 25 kPa ppH₂S, 75 kPa ppCO₂, pH 4.0, 25 °C, 1 bar, and 100 g/L NaCl).

Figure 11:

DCB specimen loaded by double tapered wedges during the K_ISSC tests, and macroscopic fracture of the DCB specimen used to measure K_ISSC in the aggressive environment (aqueous solution containing 25 kPa ppH₂S, 75 kPa ppCO₂, pH 4.0, 25 °C, 1 bar, and 100 g/L NaCl).

To validate the proposed model, four different C(T)-like notched specimens were tested in the same environment used for the S_SSC and K_ISSC tests. The maximum stress induced at the notch tip was σ_max = 1.70⋅S_SSC = 0.95⋅S_Y = 785 MPa, to keep LE conditions. These specimens’ width was w = 25 mm, and they had two initial notch sizes a/w = 0.25 and 0.33, each one with two different notch tip radii, ρ = 0.2 and ρ = 0.5 mm, see Figure 12. They were loaded by properly calibrated proof rings, after being assembled within an inert chamber, to avoid contamination of the aggressive environment. Moreover, these specimens were electrically insulated from the grips, to avoid galvanic corrosion.

Figure 12:

Notched UNS S41426 C(T)-like specimens, with four different combinations of tip radius and notch size (left). To verify the short crack tolerance under SSC predicted by the model proposed here, the specimens were first mounted inside the solution chamber (center), and then immersed into the aggressive solution loaded by proof rings, with the whole set assembled into an inert chamber, to avoid contamination during the month-long test (right).

The test period was defined considering the kinetics of the incubation time observed during the S_SSC tests, where all specimens lasted no longer than 15 days under tensile stresses higher than S_SSC. Therefore, to characterize non-propagating cracks clearly, the notched specimens were tested during 30 days inside the aggressive solution, following standard NACE TM0177 method D procedures for stainless steels. As predicted before, all these specimens cracked at the notch tip under local stresses σ > S_SSC, but none of them broke under such high loads after the 30-day immersion period.

It is important to emphasize that this tolerance to short cracks was predicted before the tests, by using the principles outlined in Sections 2 and 3 while properly calculating (instead of estimating) the SIFs of the short cracks expected at the notch tips, using mechanical tools described in the following.

Moreover, notice that since no non-propagating cracks were induced during the S_SCC tests, it is appropriate to claim that the probability of inducing such cracks by chance alone in four different notch tips with various stress concentration factors is negligible. Therefore, it can be claimed as well that these tests confirm the existence of a notch sensitivity effect in SSC, which can be quantified by proper stress analysis techniques. Therefore, structural design and analysis procedures that consider this effect do have the potential to substitute the simplistic pass-non pass criteria still very much used in such cases, possibly with significant economic gains, especially in the oil and gas industries.

5 Calculation of the SIFs of short cracks that depart from notch tips

The estimate for the SIF of short cracks that depart from notch tips presented in Section 3 can be improved, an important task in practical applications, for two reasons. First, short crack behavior is very much affected by the stress gradient ahead of the notch tip, hence imprecise estimates for its interaction with the growing (small) crack can much affect the resulting SIF value. Second, precise numerical simulations for the short crack behavior are not trivial, although recent proposals for facilitating them may be helpful (Miranda et al. 2019).

Besides the Inglis-based estimate for SIFs of short cracks that depart from notch tips used in Section 3, the Creager and Paris (1967) (C&P) estimate for stress concentration factors, based on SIFs of (long) cracks geometrically similar to the notch in question, can be used as an alternative to approximate the short crack SIFs. The idea is to use its displaced crack tip stress field as an estimate for the stress field ahead of the notch tip. Another probably better short crack SIF estimate can be obtained by using the actual stress field ahead of the notch tip, which can be easily calculated by traditional finite element (FE) procedures, instead of the Inglis or C&P approximations for it.

Notice, however, that although all such methods are quite reasonable, they cannot yield precise SIF values because they do not consider stress redistribution issues ahead of the notch tip induced by the growing (short) crack. The only way to model them is by incrementally recalculating the SIF as the crack grows. This requires non-trivial FE techniques, such as FE with highly variable sizes (which shape needs to be adjusted properly to avoid numerical instability), sequential remeshings of the entire notch region at every calculation step, and special quarter-point rosette FEs to simulate the crack tip singularity. All these problems are beyond the scope of this work, but they are properly studied in (Miranda et al. 2012).

Anyway, instead of using reasonable, but nevertheless approximate engineering estimates, FE calculations using the especially developed QUEBRA software (Miranda et al. 2003, 2012) were used to obtain precise values for the SIFs of the short cracks that started at the tips of the four notched specimens used in this work. Comparisons between short-crack tolerance predictions based on such precise FE calculations and on simpler FE and C&P estimates for those SIFs are presented in Figure 13. This analysis indicates that, at least in the studied case (but probably in similar problems as well), the proposed estimates can provide quite reasonable ball-park figures for the short crack SIFs, if precise FE calculations are not available for them. This way, they probably can be a useful engineering tool to evaluate if the predicted notch sensitivity by short crack tolerance can be a viable option to traditional pass/fail criteria, especially in practical SCC and similar EAC structural integrity analyses.

Figure 13:

K _I and K_ISSC curves versus crack size comparing C&P, FEM stress field and QUEBRA predictions (left), and similar K_I/K_ISSC curves for a 0.2 mm notch radius and 0.25 a/w relation (right).

Figure 13 indicates that, under its loading conditions, the SIF curve K_I(a) of the short crack that starts at the notch tip begins with a larger value than the crack size-dependent crack growth threshold-under-SSC curve K_ISSC(a). The point at which the K_I(a) SIF curve crosses the K_ISSC(a) threshold curve indicates where the initially propagating short crack should arrest, and the size of the resulting non-propagating crack.

Figure 13 is of course load and geometry dependent, since the K_I(a) SIF curves depend on both. For higher loads than the one used in Figure 13, the K_I(a) SIF curves would move upward and eventually would not cross the K_ISSC(a) threshold curve, indicating that the cracks initiated under SCC in that notch tip would not become non-propagating. In such cases, these cracks would not be tolerable. On the other hand, cracks initiated at lower loads than the one used in Figure 13 would stop at smaller lengths than the ones depicted in that figure. Notice that the crack driving force K_I(a) initially grows fast because the notch-concentrated stresses near the crack tip are high, but its fast increase eventually slows down due to the notch-induced stress gradient effect at the crack tip location, as discussed in Section 3. Since the sharper the notch is, the higher are its stress concentration effect and its gradient as well, see Figure 3, sharp notches are more prone to induce non-propagating cracks, whereas such cracks should not initiate from smooth surfaces in uniform stress fields. That is why cracks initiated in S_SSC tests performed in smooth specimens never become non-propagating. Indeed, eventual non-propagating cracks initiated at smooth surfaces under uniform (e.g. purely tensile) applied stresses should thus indicate a significant and variable residual stress field near their starting points.

Even though the precise calculation of short-crack K_I(a) SIF-values requires some numerical expertise and can be laborious, this mechanical reasoning is clear and straightforward, thus it should not be neglected in practical applications. However, it is fair to point out that the SIF calculation for short cracks that depart from notch tips in complex geometries in general need three-dimensional modeling techniques, but this topic is outside the scope of this work.

It is important to point out as well that the mechanics used in this work assumes isotropic and homogeneous materials. Thus, strictly speaking, its predictions should only apply for cracks larger than the grain size in metallic alloys. Those details are important, and should be explored better in future publications, but they do not invalidate the main conclusions drawn here. Indeed, like in previous works performed under liquid metal (Castro et al. 2015a) and hydrogen embrittlement (Castro et al. 2015b) EAC conditions, the simple mechanics used here was able to predict the generation of non-propagating cracks from notch tips loaded under local stresses σ >> S_SCC, induced by still another EAC mechanism (SSC in this case) inside an aggressive environment.

Figure 14 confirm this claim. It was taken from two C(T)-like notched specimens with ρ = 0.2 mm tip radius after a 30-day immersion, one with a/w = 0.25 and the other with a/w = 0.33 crack-to-width ratios. It is important to emphasize that, even under a maximum local stress at the notch tips much higher than S_SCC, namely σ_max = 1.70⋅S_SSC = 0.95⋅S_Y, the specimens did not break after immersed for 30 days in the aggressive solution. Moreover, notice that, due to the highly localized stress level, several small non-propagating cracks start from the notch tip, and that the largest non-propagating crack sizes are similar to those predicted in Figure 13. Since the model used to predict them has no data-fitting parameters, these experimental results clearly indicate that the proposed mechanical model indeed may be a potentially useful tool to design and analyze structural components used in SSC conditions, especially for the oil and gas industries.

Figure 14:

UNS S41426 steel specimen with a_i/w = 0.25 and 0.33, ρ = 0.2 mm, after 30 days immersed in medium containing 250 mbar ppH₂S, 750 mbar ppCO₂, pH4.0, 25 °C, 1 bar, 100 g/L NaCl.

Figure 15 shows the SEM analysis around the crack from the C(T)-like notched specimens with ρ = 0.2 mm tip radius, with a/w = 0.25, after 30 days of immersion. The images confirm the mixed-mode cracking mechanism with transgranular and intergranular cracking, as Figure 10 previously indicated from S_SSC tests. The steel matrix indicates the presence of some cubic precipitates of Titanium Nitride (TiN) and other smaller and spherical precipitates of Titanium Carbides (TiC).

Figure 15:

Region around the largest crack found on a UNS S41426 steel specimen with a_i/w = 0.25 and ρ = 0.2 mm, after 30 days immersed in medium containing 250 mbar ppH₂S, 750 mbar ppCO₂, pH4.0, 25 °C, 1 bar, 100 g/L NaCl.

6 Conclusions

An S13Cr supermartensitic stainless steel was tested under sulfide stress corrosion (SSC) cracking conditions inside an aggressive sulfide environment (pH 4, 100 g/L of Cl⁻ in water solution, 1 atm total pressure, and 25.8 kPa (3.75 psi) of H₂S with CO₂ in balance). Its resistances to crack initiation S_SSC and growth K_ISSC were measured by standard procedures. Then, four different C(T)-like notched specimens were designed to simulate a peak stress of 0.95⋅S_Y >> S_SCC at the notch tip, to ensure that short SSC cracks would start there under nominally linear elastic conditions, but would stop after growing for a while, becoming non-propagating cracks. The experimental results confirmed the notch sensitivity under SSC predictions, just as shown before for other EAC mechanisms. These results support the claim that sound stress analysis principles should be considered as an option to over-conservative pass/fail criteria when selecting materials for EAC service.